The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an important place in cryptology. Finite field, also known as Galois Fields, are cornerstones for understanding any cryptography. This encryption method on AES is a method that uses polynomials on Galois fields. In this paper, we generalize the AES-like cryptology on 2 × 2 matrices. We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm. So, this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.

In this study, we describe the generalized k-order Fibonacci and Lucas numbers and give some important results with specific choices. The main purpose of this study is to define the generalized k-order Fibonacci hybrid and Lucas hybrid numbers and give some of their properties. With this definition, we obtain some important number sequences by making special choices. For example, we give the Horadam hybrid number for k=2 and the other series of hybrid numbers, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal end Jacobsthal-Lucas hybrid numbers. We give the generating functions and some properties about the generalized k-order Fibonacci hybrid and Lucas hybrid numbers. Also, we identify and prove the matrix representation for the generalized k-order Fibonacci hybrid and Lucas hybrid numbers. The Qk matrix given for k-order Fibonacci numbers are defined for the generalized k-order Fibonacci hybrid and Lucas hybrid numbers and after the matrices with the generalized k-order Fibonacci hybrid and Lucas hybrid numbers are obtained with help of auxiliary matrices, important relationships and identities are established.

In this paper we define k-order Gaussian Fibonacci polynomials with boundary conditions and give the generating function, explicit formula and some identities for k-order Gaussian Fibonacci polynomials. We introduce the matrix represent and we obtain the k-order Gaussian Fibonacci Polynomials matrix. We define a new coding theory called k-order Gaussian Fibonacci Polynomials coding theory and establish the code elements for values of k. This coding/decoding method bound to the Q_k (x), R_k (x) and E_k,n (x) matrices. So, this method is different from the classical algebraic coding. Consequently, with this method, we move the coding theory onto a complex space which is a different field. Therefore, new working areas are created.

In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged by taking \begin{document}$ x = 1 $\end{document}. We call this coding theory the k-order Gaussian Fibonacci coding theory. This coding method is based on the \begin{document}$ {Q_k}, {R_k} $\end{document} and \begin{document}$ E_n^{(k)} $\end{document} matrices. In this respect, it differs from the classical encryption method. Unlike classical algebraic coding methods, this method theoretically allows for the correction of matrix elements that can be infinite integers. Error detection criterion is examined for the case of \begin{document}$ k = 2 $\end{document} and this method is generalized to \begin{document}$ k $\end{document} and error correction method is given. In the simplest case, for \begin{document}$ k = 2 $\end{document}, the correct capability of the method is essentially equal to 93.33%, exceeding all well-known correction codes. It appears that for a sufficiently large value of \begin{document}$ k $\end{document}, the probability of decoding error is almost zero.

In this paper, we define and study another interesting generalization of the Fibonacci quaternions is called k-order Fibonacci quaternions. Then we obtain for Fibonacci quaternions, for Tribonacci quaternions and for Tetranacci quaternions. We give generating function, the summation formula and some properties about k-order Fibonacci quaternions. Also, we identify and prove the matrix representation for k-order Fibonacci quaternions. The matrix given for k-order Fibonacci numbers is defined for k-order Fibonacci quaternions and after the matrices with the k-order Fibonacci quaternions is obtained with help of auxiliary matrices, important relationships and identities are established.

In this paper we introduce and study k-order Gaussian Fibonacci Coding Theory. We give illustrative examples about coding theory. This coding theory is a method bound to the Q_k, R_k and E_n^(k) matrices. This coding/decoding method is different from classical algebraic coding. k-order Gaussian Fibonacci Coding method depends on matrix multiplication and can be performed quickly and easily by today’s computers. This method will not only ensures information security in data transfer but also has high correct ability. Consequently, this method aims to increase the reliability of information transfer by moving the coding theory to the complex space.